Multiple Periodic Solutions of Allen-Cahn System Involving Fractional Laplacian

Abstract

We consider periodic solutions of the following nonlinear system associated with the fractional Laplacian

$ \left(-\partial_{x x}\right)^{s} \boldsymbol{u}(x)+\nabla F(\boldsymbol{u}(x))=0 \quad \text { in } \mathbb{R},$

where F : $ \mathbb{R}^{2} \rightarrow \mathbb{R}$ is a smooth double-well potential. For the case that F is even in its two variables we obtain the existence of more and more periodic solutions with large period, by using Clark’s theorem. For the case that F is only even in its the second variable and the origin is a saddle critical point of F, we give two periodic solutions by using Morse theory.

Keywords

Multiple periodic solutions / fractional Laplacian / Allen-Cahn system / Clark’s theorem / Morse theory